Mathematics

# Solve $\int {\sqrt {\dfrac{{1 + x}}{{1 - x}}} }$

##### SOLUTION

$\int {\sqrt {{{1 + x} \over {1 - x}}} dx}$

$put\,x = \cos 2\theta$

$d\theta = 12{\sin ^2}\theta \,d\theta$

$\int {\sqrt {{{1 + {{\cos }^2}\theta } \over {1 - {{\cos }^2}\theta }}} \times 2{{\sin }^2}\theta \,d\theta }$

$\int {\sqrt {{{2{{\cos }^2}\theta } \over {2{{\sin }^2}\theta }}} \times - 2{{\sin }^2}\theta \,d\theta }$

$- 2\int {{{\cos \theta } \over {\sin \theta }} \times 2\sin \theta \cos \theta \,d\theta }$

$- 4\int {{{1 + {{\cos }^2}\theta } \over 2}d\theta }$

$- 2\left( {\theta + {{{{\sin }^2}\theta } \over 2}} \right) + C$

$- 2\theta - {\sin ^2}\theta + C$

$- {\cos ^1}x - \sqrt {1 - {x^2}} + c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Single Correct Medium
Evaluate the following integration w.r.t. $x$
$x^{2}\int (1-\dfrac {2}{x})^{2}dx$
• A. $\left[\dfrac{x^3}{3}+4x\right]+c$
• B. ${\sqrt2}+c$
• C. $\left[\dfrac{x^3}{2}\right]+c$
• D. $\left[\dfrac{x^3}{3}+4x+\dfrac{4}{x^2}\right]+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Evaluate $\displaystyle \int xsecx.tanxdx=$
• A. $x\sec x+\log|\tan(\pi/2+x/2)|+c$
• B. $x\sec x-\log |\tan(\pi/4+x)|+c$
• C. $x\sec x+\log|\tan x/2|+c$
• D. $x\sec x-\log$$|\tan(\pi/4+x/2)|+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove that $\displaystyle\int^{\pi/2}_0\dfrac{\cos^3xdx}{(\sin^3x+\cos^3x)}=\dfrac{\pi}{4}$.

1 Verified Answer | Published on 17th 09, 2020

Q4 One Word Medium
$\displaystyle I= \int_{0}^{\pi /2}\frac{x\sin x\cos x}{\cos ^{4}x+\sin ^{4}x}dx$
$\displaystyle \therefore I= \pi ^{2}/k.$
what is k?

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$